An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

127

18. Determine whether the function zu=x3+ay2-bxy+cx,
admits of a maximum or a minimum, and find its value.
19. Given u ==/(x-a)2+y2+b2 + /(x-a)2+y-/32, to
find whether u admits of a maximum or a minimum, and
when possible, to determine its value.
20. Determine the different maximum or minimum values
of the function u = x' + y2 + b + Va2 - (x - c)2 Y, and apply
the criterion to decide which they are.
21. Find what maximum or minimum values the function u = x4 + y4 - ax2y - axy2 + c2x2 + c2y2 admits of, and
decide which they are.
22. In the equation u = a sin x -+b sin y - c sin (x - y),
it is required to deduce the equations by means of which a
maximum or minimum value of u may be ascertained.
23. If u-=xy /a'b2" —a2 - by2", it is required to find
its maximum or minimum value, and to apply the criterion
for deciding which it is.
24. If u = a xy2z3- -_ Xy23 - xy3z -_ xy2z4 it is required
to prove that when 'x I= a y = a and a =3, the value of
7
fa\
u= -, a maximum.
\77
25. Given u =ax2y3z4 —x3y z4 _ —,2y4z4-x 2y35, it is
required to prove that when x=aa, y=-a and z=2a, the
value of u = a maximum.
26. It is required to prove that the function
u = a - bxy + cxz + y,
admits of neither a maximum nor a minimum.
27. Required the values of x, y and z, which shall render
xyz
the function u = -xy a maximum
(a + x) (x + y) (y + z) (z+ b)
or a minimum.